Chapter 5: Q. 81 (page 479)

Suppose f(x) is continuous on R and that for some real number c, both
$\int_{- \infty}^{C} f (x) d x + \int_{c}^{\infty} f (x) d x$ exist. Use properties of definite integrals to prove that for all real numbers d, $\int_{- \infty}^{C} f (x) d x + \int_{c}^{\infty} f (x) d x$ is equal to $\int_{- \infty}^{d} f (x) d x + \int_{d}^{\infty} f (x) d x .$

Short Answer

Expert verified

The given statement is proved.

Step by step solution

Step 1. Given Information.

The given integral is $\int_{- \infty}^{C} f (x) d x + \int_{c}^{\infty} f (x) d x .$

Step 2. Prove.

To prove that for all real numbers d, $\int_{- \infty}^{C} f (x) d x + \int_{c}^{\infty} f (x) d x$ is equal to $\int_{- \infty}^{d} f (x) d x + \int_{d}^{\infty} f (x) d x .$ Let the function f is continuous on [a,b] and for any real number c, $\int_{a}^{b} f (x) d x = \int_{a}^{C} f (x) d x + \int_{c}^{b} f (x) d x .$

We will prove the given statements in two cases when c < d and whend < c.

Step 3. Estimate when c < d.

We will use the property of definite integrals,

$\int_{- \infty}^{C} f (x) d x + \int_{c}^{\infty} f (x) d x = \int_{- \infty}^{C} f (x) d x + \int_{c}^{d} f (x) d x + \int_{d}^{\infty} f (x) d x Use \int_{a}^{b} f (x) dx = \int_{a}^{C} f (x) dx + \int_{c}^{b} f (x) dx = \int_{- \infty}^{C} f (x) dx + \int_{c}^{\infty} f (x) dx$

Step 4. Estimate when d < c.

We will use the property of definite integrals,

$\int_{- \infty}^{d} f (x) d x + \int_{d}^{\infty} f (x) d x = \int_{- \infty}^{d} f (x) d x + \int_{d}^{c} f (x) d x + \int_{c}^{\infty} f (x) d x Use \int_{a}^{b} f (x) dx = \int_{a}^{c} f (x) dx + \int_{c}^{b} f (x) dx = \int_{- \infty}^{C} f (x) dx + \int_{c}^{\infty} f (x) dx$

Thus, role="math" localid="1649143526896" $\int_{- \infty}^{c} f (x) d x + \int_{c}^{\infty} f (x) d x = \int_{- \infty}^{d} f (x) d x + \int_{d}^{\infty} f (x) d x .$

Hence proved.

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove.

Step 3. Estimate when c < d.

Step 4. Estimate when d < c.

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Suppose f(x) is continuous on R and that for some real number c, both ∫-∞Cf(x)dx+∫c∞f(x)dx exist. Use properties of definite integrals to prove that for all real numbers d,∫-∞Cf(x)dx+∫c∞f(x)dxis equal to∫-∞df(x)dx+∫d∞f(x)dx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove.

Step 3. Estimate when c < d.

Step 4. Estimate when d < c.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Mechanics Maths

Discrete Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.